Seminars & Colloquia Calendar

Abstract:  Cluster varieties are a particularly nice class of log Calabi-Yau varieties the non-compact analogue of usual Calabi-Yaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural progression from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations. Compactifications of A and their toric degenerations were studied extensively by Gross, Hacking, Keel, and Kontsevich. These compactifications generalize the polytope construction of toric varieties a construction which is recovered in the central fiber of the degeneration. Compactifications of X were introduced by Fock and Goncharov and generalize the fan construction of toric varieties. Recently, Lara Bossinger, Juan Bosco Frías Medina, Alfredo Nájera Chávez, and I introduced the notion of an X-variety with coefficients, expanded upon the notion of compactified X-varieties, and for each torus in the atlas gave a toric degeneration where each fiber is a compactified X-variety with coefficients. We showed that these fibers are stratified, and each stratum is a union of compactified X-varieties with coefficients. In the central fiber, we recover the toric variety associated to the fan in question, and we show that strata of the fibers degenerate to toric strata.

This talk is based on arXiv:1809.08369 [math.AG].

November 8